panel data unit roots and cointegration: an overview

PANEL DATA UNIT ROOTS AND COINTEGRATION:AN OVERVIEW

Anindya Banerjee�

I. INTRODUCTION

The analysis of unit roots and cointegration in panel data has been a fruitfularea of study in recent years, with Levin and Lin (1992, 1993) and Quah(1994) being the seminal contributions in this ®eld. The investigation ofintegrated series in a panel data context is based on two separate but richlydeveloped ®elds of econometric investigation, the ®rst being unit roots andcointegration in time series and panel data econometrics the other. Bothliteratures have been surveyed in rich detail. For unit roots and cointegrationin time series, Banerjee et al. (1993), Hamilton (1994) and Phillips andXiao (1998) are useful sources. The book by Baltagi (1995) and the editedcollection by Matyas and Sevestre (1996) are important references for theliterature on panel data econometrics. The several volumes of papers editedby Maddala (1994) are also relevant in this regard.

The emphasis of the literature on unit roots and cointegration in paneldata has been the attempt to combine information from the time seriesdimension with that obtained from the cross-sectional, in the hope thatinference about the existence of unit roots and cointegration can be mademore straightforward and precise by taking account of the cross-sectiondimension, especially in environments in which the time series for the datamay not be very long but very similar data may be available across a cross-section of units such as countries, regions, ®rms or industries.

The empirical motivations have therefore always been important. Further-more, with increasingly larger quantities of panel data information becom-ing available, the investing of effort in this area of research has seemedworthwhile, given the well-known power de®ciencies of pure time series-

OXFORD BULLETIN OF ECONOMICS AND STATISTICS, SPECIAL ISSUE (1999)0305-9049

607# Blackwell Publishers Ltd, 1999. Published by Blackwell Publishers, 108 Cowley Road, Oxford

OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA.

�Paper prepared as editorial introduction for the special issue of the Oxford Bulletin ofEconomics and Statistics. I am obliged to the contributors to this volume for their comments andto Wallace Lo for valuable research assistance. A substantial part of the work for this paper andfor putting together the special issue was undertaken by me during March and April 1999 in theDepartment of Economics at the University of Canterbury in Christchurch, New Zealand, underthe auspices of a Visiting Erskine Fellowship. The generosity of the Erskine Foundation and theDepartment of Economics in Canterbury is gratefully acknowledged. In particular, I thank RobinHarrison and Alfred Haug for their hospitality. I also thank the ESRC for funding this researchunder grant L116251015.

based tests for unit roots and cointegration. Panel data techniques have beenused to investigate, for example, wages in unionized and non-unionizedindustries (Breitung and Mayer (1994)), purchasing power parity (Bernardand Jones (1996), Coakley and Fuertes (1997), Frankel and Rose (1996),MacDonald (1996), O'Connell (1998), Oh (1996), Pedroni (1995, 1997a),Papell (1997), Wu (1996)) and issues of convergence (Cechetti et al.(1999), Evans and Karras (1996), Lee, Pesaran and Smith (1997), Pedroni(1998)). The papers by Suzanne McCoskey and Chihwa Kao and by ChihwaKao, Min-Hsien Chiang and Bangtian Chen in this Special Issue look aturbanization and international R&D spillovers respectively and are de-scribed later in the introduction.

The literature that has emerged has liberally drawn many elements fromits parent literatures. The consideration of fully modi®ed estimation techni-ques, to take account of endogeneity of the regressors and correlation andheteroscedasticity properties of the residuals, on the one hand, and the useof methods for ®xed or random effects estimation, developed in theliterature on panel data with stationary variables, on the other, are twoexamples of where the clear links may be identi®ed. Yet, as in otherinstances where a new literature comes to be seen to be signi®cant, theaggregate has turned out to be greater than the sum of its parts and thetheory and practice of integrated series in panel data have given rise to a setof interesting and surprising results which are uniquely its own.

A few examples of the distinctive features will suf®ce to elaborate on thepoint made in the previous paragraph. From the early papers which devel-oped the asymptotic theory of unit root processes in time series (Phillips(1987), Engle and Granger (1987)), many of the estimators and statistics ofinterest have been shown to have limiting distributions which are compli-cated functionals of Wiener processes. In direct contrast, the asymptotics ofnon-stationary panels, starting with Levin and Lin, have led to demonstra-tions of estimators having Gaussian distributions in the limit. These resultshave been extended to allow for a wide degree of heterogeneity across theunits comprising the panel.

The limiting distributions have also required the development and use ofmultivariate `panel' functional central limit theorems, since the limitingbehaviour has required consideration of processes indexed by not only timebut also by unit. The formal and general treatment of the asymptoticbehaviour of such double indexed integrated processes has begun onlyrecently (Phillips and Moon (1999a)), although various aspects had beenimplicitly employed in the earlier literature. It has become clear that severalapproaches are possible and the limit of the processes may depend on theassumptions made about the manner in which N (the units) and T tend toin®nity. For example, one may ®x N and let the other index pass to in®nityand subsequently allow N to tend to in®nity. This is denoted by Phillips andMoon (1999a) as (N , T !1)seq. Alternatively N and T may be allowed topass to in®nity at a controlled rate of the type T � T (N ). A third possibility

608 BULLETIN

# Blackwell Publishers 1999

is for both indexes to pass to in®nity simultaneously without any restrictionsbeing placed on the divergence denoted by Phillips and Moon as(N , T !1). These are termed sequential, diagonal path and joint limitsrespectively. Examples of the use of all three forms of limiting behaviourare available in the literature and one of the contributions of the newasymptotics has been to develop wherever possible the equivalences orrelations between the various modes of convergence, possibly under thespeci®cation of restrictions (see Phillips and Moon (1999a)).

Spurious regressions in panels have also been shown to have interestingproperties. Phillips and Moon (1999a) demonstrate how panel methodsallow for the estimation of a long-run relation among variables even incases where consideration of the time dimension alone would lead to theregression being characterized as spurious. Thus in such cases the strongnoise in the time series regression is attenuated by pooling the cross-sectionand time series observations. The meaning of this so-called long-runrelation is however open to some interpretation and, as noted in Theorem 1of Kao (1999), the t-statistic of the estimator is divergent as in pure timeseries models.

The case for the importance and distinctiveness of this literature istherefore well established. The techniques will be developed and put tomany more varied and interesting uses in the future and this Special Issuebrings together contributions from some of the leading researchers in this®eld. The space available in the introduction does not allow for more than aselective consideration of the literature. McCoskey and Kao (1998b) andPhillips and Moon (1999b) are useful overviews. Our aim here is four-fold:®rst, to provide what might be called a catalogue raisonneÂ of the maintechniques and papers and thereby to take stock of the literature; second, topresent a uni®ed account of the main themes which link the research in thisarea; third, to place the papers in this volume within the context of theliterature; and fourth, to put forward new areas for research. The discussionis divided into two main sections consisting of testing for unit roots andtesting for cointegration. Among papers in the former category are Im,Pesaran and Shin (1997), Kao (1999), Levin and Lin (1992, 1993), Quah(1994) and the papers by G. S. Maddala and Shaowen Wu and by HyungsikMoon and Peter Phillips in the Special Issue, while in the latter belong Kaoand Chiang (1998), Pedroni (1995, 1996, 1997a), McCoskey and Kao(1998a), Phillips and Moon (1999a) and the contributions by Peter Pedroniand Stephen Hall, Stepana Lazarova and Giovanni Urga printed here.

Two further papers are worth mentioning here since they have animportant bearing on this literature and are the motivator of the Hall et al.analysis in this volume. Robertson and Symons (1992) and Pesaran andSmith (1995) (extended in Im, Pesaran and Smith (1996)) were veryin¯uential in demonstrating the inconsistency of estimators in dynamicheterogeneous panels which use pooled or aggregated data and recom-mended the use of group-mean estimators. The latter is the basis of the unit

PANEL DATA UNIT ROOTS AND COINTEGRATION 609


root test proposed in Im et al. (1997). Hall et al. provide an interesting andimportant counter-example to the analysis by Pesaran and his co-authorswhile at the same time moving the literature on to a more general direction.The Phillips and Moon long-run relation alluded to above is also anotherexample of how looking at integrated data in panels led to some widely heldbeliefs being modi®ed.

The next section describes testing for unit roots, using the analysis inLevin and Lin (1992, 1993) as its starting point. Section 2.2 provides anaccount of the Im et al. (1997) tests and Section 2.3 introduces the Maddalaand Wu paper in this volume. Section 3 moves on to the discussion of testsfor cointegration, with Section 3.1 presenting Pedroni's tests for cointegra-tion. The construction of these tests is described in detail in Pedroni's paper.Section 3.2 analyses the LM-test for cointegration developed by McCoskeyand Kao (1998a) and describes an empirical application of the methodcontained in the Special-Issue paper by McCoskey and Kao. This sectionalso introduces the paper by Kao et al. Section 4 discusses new directionsfor research. In particular, we discuss Phillips and Moon's formalization ofthe asymptotic theory of integrated panel data. The topic of their contribu-tion to the Special Issue is the estimation of local-to-unity parameters in thepresence of homogeneous or heterogeneous deterministic trends, and theirpaper provides a new and important example of inconsistent maximum-likelihood-estimation in dynamic panels. Section 5 concludes. The notation

used throughout the volume is standard and self-explanatory. �)d or )denotes convergence in distribution while ! p or ! is used for conver-gence in probability The non-stochastic limit (in®nite or ®nite) of asequence is also denoted by ! and the context makes the usage clear. InWiener integrals of the form

�W (r)dr, the argument r is often suppressed.

II. TESTING FOR UNIT ROOTS IN DYNAMIC PANELS

2.1. Levin and Lin

The structure of the Levin and Lin analysis may be summarized in thefollowing equation:

Äyi, t � ái � äi t � èt � ri yi, tÿ1 � æi, t, i � 1, 2, . . . , N , t � 1, 2, . . . , T :

(1)

It therefore allows for ®xed effects and unit-speci®c time trends inaddition to common time effects (which may in practice be concentratedout of the equation). The unit-speci®c ®xed effects are an important sourceof heterogeneity here since the coef®cient of the lagged dependent variableis restricted to be homogeneous across all units of the panel. The Levin andLin tests amount to testing for the null hypothesis H0 : ri � 0 for all iagainst the alternative H A : ri � r, 0 for all i, with auxiliary assumptions

610 BULLETIN


under the null also being required about the coef®cients relating to thedeterministic components.

The main theorems in Levin and Lin relate to deriving the asymptoticdistributions of the panel estimator of r under different assumptions on theexistence of ®xed effects or heterogeneous time trends. The simplest casesto consider are for æi, t � IID(0, ó 2) for ®xed i. The errors are also assumedto be independent across the units of the sample.1 For example, ifái � äi � 0 for all i and there are no common time effects, then theasymptotic distribution of the ordinary least squares (OLS) pooled panelestimator r is given by

T��Np

r) N (0, 2), T , N !1 (2)

tr�0 ) N (0, 1): (3)

Here the convergence rate to normality of the coef®cient estimator isfaster as T !1 than as N !1. For a more general model, say with nocommon time effects and äi � 0, the relevant distributional result is givenby,

T��Np

r� 3��Np) N(0, 10:2),

��Np

=T ! 0, T , N !1 (4)��1:25p

:tr�0 ��1:875Np

) N(0, 645=112),��Np

=T ! 0, T , N !1: (5)

The null hypothesis for this second model is given by H0 : r � 0, ái � 0for all i against the alternative H0 : r, 0, ái unrestricted. The signi®canceof the Levin and Lin analysis was the ®rst formal demonstration of thecorrection and standardization factors required in order for the unit rootestimators to have Gaussian distributions in the limit. Some of the resultswere conditional upon a particular controlled rate of divergence of N and Tto in®nity, so that

��Np

=T ! 0 and hence the time dimension expands moreslowly than the cross-section. The scaling (by powers of both N and T ) isalso noteworthy in contrast with the scaling factor appropriate for theanalogous Dickey±Fuller estimator in time series, where a single power ofT would be suf®cient.

The expressions such as 3��Np

or��1:875Np

are simple examples ofcentring corrections needed in order for the statistics to have mean zeroasymptotically. The variance standardizations necessary are also easilydeduced. The important extension undertaken by Levin and Lin (1993) wasto look at the case where the error processes had more general correlatedand heteroscedastic structures, although the independence across cross-section units was retained. In the context of dynamic models such as thosegiven above, the introduction of, say, serial correlation in the residuals has

1The condition that the individual processes are cross-sectionally independent is a criticalassumption in much of the literature. Later on we discuss relaxations of this condition but unlessotherwise stated it should be taken as given.



important consequences by introducing correlation between the laggeddependent variable and the error. Along with the introduction of hetero-scedasticity, the use of corrections in order to obtain statistics free fromnuisance parameters is thereby necessitated.

Consider now the second of the two models discussed above where theerror process now follows a stationary invertible ARMA process for eachunit and is distributed independently across units. Thus,

æi, t �X1j�0

èijæi, tÿ j � åi, t: (6)

For all i and t, æi, t has ®nite non-zero fourth moments, the variance of theinnovation process åi, t is bounded from below away from zero and thevariation at frequency zero (long-run variance) of æi, t is bounded above.

Levin and Lin (1993) prescribe the use of augmented Dickey±Fuller(ADF) test to each individual series as the starting point of their testingprocedure for unit roots. Thus, the regressions (for each i)

Äyi, t � ri yi, tÿ1 �Xpi

j�1

èijÄyi, tÿ j � ái � åi, t, t � 1, 2, . . . , T (7)

is estimated by regressing ®rst Äyi, t and then yi, tÿ1 on the remainingvariables in (7), providing the residuals ei, t and Vi, tÿ1 respectively. Theregression of ei, t on Vi, tÿ1

ei, t � ri Vi, tÿ1 � åi, t (8)

is then estimated to derive ri from the ith cross-section. The followingexpressions are next required:

ó 2ei� (T ÿ pi ÿ 1)ÿ1

XT

t� pi�2

(ei, t ÿ ri Vi, tÿ1)2

~ei, t � ei, t=ó ei

~Vi, tÿ1 � Vi, tÿ1=ó ei

ó 2yi� (T ÿ 1)ÿ1

XT

t�2

Äy2i, t � 2

XK

L�1

wK L (T ÿ 1)ÿ1XT

t�L�2

Äyi, tÄyi, tÿL

!2

SN ,T � Nÿ1XN

i�1

(ó yi=ó ei

):

2This is an estimate of the long-run variance of yi. K is the lag truncation parameter and wK L isthe lag window.

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The ®nal step is to estimate the panel regression (making use of all i and t)

~ei, t � r~Vi, tÿ1 � ~åi, t (9)

and compute the t-statistic

tr�0 � rRSE(r)

, (10)

where

RSE(r) � ó åXN

i�1

XT

t� pi�2

~V 2i, tÿ1

24 35ÿ1=2

ó 2å � (N ~T)ÿ1

XN

i�1

XT

t� pi�2

(~ei, t ÿ r~Vi, tÿ1)2

p � Nÿ1XN

i�1

pi, ~T � (T ÿ pÿ 1):

The Levin and Lin statistic is an adjusted version of (10) above. It isgiven by

tr� � tr�0 ÿ N ~T SN ,T óÿ2å RSE(r)ì~T

ó ~T

, (11)

where ì~T and ó ~T are mean and standard deviation adjustment terms whichare computed by Monte Carlo simulation and tabulated in their paper forthree separate speci®cations of the deterministic terms in (1) above. Theadjustment terms are available for a given regression model, time seriesdimension ~T ranging from 25 to 250 and lag truncation parameter Kvarying from 9 to 20. N (the cross-section dimension) is set to 250 in all theadjustment term simulations.3 The central theorem of the Levin and Linanalysis showed that provided the ADF lag order pmax increased at somerate T p where 0 , p < 1=4 and the lag truncation parameter K increased atrate T q where 0 , q , 1, then under the null hypothesis that r � 0, thepanel test statistic tr� has the property that as T , N !1,

tr� ) N(0, 1): (12)

3An element of the discussion in many of the papers where such corrections are proposed is topresent them for ®nite T , N and also for so-called asymptotic or large T , N and to see how wellthe asymptotic values (since these are most easily usable) approximate the ®nite samplecorrections. We draw attention to this feature but do not enter into discussion of this importantarea.



Under the alternative hypothesis, tr� diverges to negative in®nity at rate��NTp

, a property that can thereby be exploited to construct a consistentone-tailed test of the null hypothesis against the alternative.

In summary, the Levin and Lin papers contained almost all the keyelements which continue to preoccupy the later discussions in the literature.These included (a) the demonstration of asymptotic normality subject tosuitable scaling and corrections such as in (2)±(5) above; (b) the necessityof focusing on the rates at which T and N tend to in®nity; (c) the issue ofhomogeneity versus heterogeneity (under the null and the alternativehypotheses). The Levin and Lin framework was restrictive in this sense inrequiring r to be homogeneous across i; (d) the maintained assumption ofindependence across cross-section units and (e) the corrections or modi®ca-tions required to allow for dependent and heteroscedastic error processesand the resulting endogeneity of the regressors.

Generalizations of corrections of the form given above are a key part ofthe tests for unit roots and cointegration discussed below and in theconstruction of fully-modi®ed estimators proposed by Pedroni (1996) andKao and Chiang (1998). Three of these elements, namely (c)±(e), are thusvery important for what follows next in our discussion. Im et al. (1997) andMaddala and Wu propose important relaxations of (c) and (d) while (e)becomes very signi®cant in the context of testing for cointegration in panelsand is dealt with in more detail in Section 3. We return to (b) in Section 4 ofthis paper when discussing the work of Phillips and his co-authors.

2.2. Im, Pesaran and Shin

The main extension by Im et al. (1997) of the Levin and Lin framework wasto allow for heterogeneity in the value of ri under the alternative hypothesis.A small modi®cation of (1) above makes the point clearly. Let:

Äyi, t � ái � ri yi, tÿ1 � æi, t, i � 1, 2, . . . , N ; t � 1, 2, . . . , T : (13)

The null and alternative hypotheses are de®ned as

H0 : ri � 0 for all i (14)

against the alternatives

H A : ri , 0, i � 1, 2, . . . , N1, ri � 0, i � N1 � 1, N1 � 2, . . . , N (15)

and the errors æi, t are serially autocorrelated (as in (6)) with different serialcorrelation (and variance) properties across units.

In view of the well-known objections of Pesaran and Smith (1995) on theuse of pooled panel estimators, such as those used by Levin and Lin (1992,1993), for processes which display heterogeneity of the kind given in (13)±(15) above, Im et al. (1997) propose the use of a group-mean Lagrangemultiplier (LM) statistic to test for the null hypothesis in (14). The ADFregressions

614 BULLETIN


Äyi, t � ri yi, tÿ1 �Xpi

j�1

èijÄyi, tÿ j � ái � åi, t, t � 1, 2, . . . , T (16)

are estimated and the LM-statistic for testing ri � 0 is computed. De®ning

LM N ,T � Nÿ1XN

i�1

LM i,T ( pi, èi), (17)

where èi � (èi1, èi2, . . . , èipi)9 and LM i,T ( pi, èi) is the individual LM

statistic for testing ri � 0, the standardized LM-bar statistic is given by

ØLM �

��Np

LM N ,T ÿ Nÿ1XN

i�1

E[LM i,T ( pi, 0)jri � 0]

( )��Nÿ1

XN

i�1

Var[LM i,T ( pi, 0)jri � 0]

s : (18)

The values of E[LM i,T ( pi, 0)jri � 0] and Var[LM i,T ( pi, 0)jri � 0] areobtainable by stochastic simulation and are tabulated in their paper using50,000 replications for different values of T and pi's. It is shown that underH0 : ri � 0 for all i,

ØLM ) N(0, 1) (19)

as T , N !1 and N=T ! k where k is a ®nite positive constant. For thetest to be consistent under the alternative, it is also required thatlimN!1(N1=N ) � ë1, 0 , ë1 < 1. Under this further assumption, ØLMdiverges to positive in®nity at rate T

��Np

under the alternative.Im et al. (1997) also propose the use of a group-mean t-bar statistic given

by

Ø t �

��Np

tN ,T ÿ Nÿ1XN

i�1

E[ti,T ( pi, 0)jri � 0]

( )��Nÿ1

XN

i�1

Var[ti,T ( pi, 0)jri � 0]

s , (20)

where

tN ,T � Nÿ1XN

i�1

ti,T ( pi, èi), (21)

and ti,T ( pi, èi) is the individual t-statistic for testing ri � 0 for all i.E[ti,T ( pi, 0)jri � 0] and Var[ti,T ( pi, 0)jri � 0] are tabulated in the paper.The convergence result stated for ØLM holds for Ø t also and consistency isguaranteed under the controlled rate of divergence of N and T to in®nitysuch that N=T ! k.



In a Monte Carlo Study, Im et al. (1997) demonstrate better ®nite sampleperformances of the ØLM and Ø t in relation to the Levin and Lin test givenby tp�.

2.3. Maddala and Wu

The focus of the Maddala and Wu paper in this Special Issue is toconcentrate on the shortcomings of both the Levin and Lin and Im et al.(1997) frameworks and to offer an alternative testing strategy. Maddala andWu argue that while the Im et al. (1997) tests relax the assumption ofhomogeneity of the root across the units, several dif®culties still remain.First, common with most other tests proposed both for unit roots andcointegration in panels, Im et al. (1997) assume in their basic frameworkthat T is the same for all the cross-section units and henceE[ti,T ( pi, 0)jri � 0] and Var[LM i,T ( pi, 0)jri � 0], for example, is the samefor all i conditional upon the lag length. Thus to consider the case ofunbalanced panels further simulations are required. Second, as with Levinand Lin, the critical values are sensitive to the choice of lag lengths in theADF regressions. Third, the results in these papers apply to only a limitedclass of tests of the unit root hypothesis in panels and do not apply to testssuch as those proposed by Elliott, Rothenberg and Stock (1996). Fourth,and most importantly, while Im et al. (1997) allow for a limited amount ofcross-correlation across units by allowing for common time effects, so thatèt in (1) is non-zero, Maddala and Wu suggest quite rightly that in manyreal-world applications the cross-correlations are unlikely to take thissimple form (which can in any case be effectively eliminated by subtractingout the cross-section mean from yi, t before applying the tests). Maddala andWu's analysis, using bootstrapping, represents the ®rst signi®cant steptowards allowing dependence of a more general form.

Maddala and Wu propose the use of a test due to Fisher (1932) which isbased on combining the p-values of the test-statistic for a unit root in eachcross-sectional unit. The Fisher test is non-parametric, and may be com-puted for any arbitrary choice of a test for the unit root. It is an exact testand the statistic given by

ÿ2XN

i�1

ln(ði) (22)

is distributed as a chi-squared variable with 2N degrees of freedom underthe assumption of cross-sectional independence. ði is the p-value of the teststatistic in unit i. The obvious simplicity of this test and its robustness tostatistic choice, lag length and sample size make it extremely attractive. Inparticular, one may use tests where either stationarity or integration is themaintained hypothesis to compute the p-values. More signi®cantly, Mad-dala and Wu demonstrate the use of bootstrapping methods to extend the

616 BULLETIN


framework of testing for unit roots in panels to allow for cross-correlations.The details are contained in their paper published here where the Fisher testwith bootstrap-based critical values is given as the preferred choice.

III. TESTING FOR COINTEGRATION IN DYNAMIC PANELS

The literature on testing for cointegration in panels has so far taken twobroad directions. The ®rst consists of taking as the null hypothesis that ofno cointegration and using residuals derived from the panel analogue of anEngle and Granger (1987) static regression to construct the test statisticsand tabulate the distributions. The most general statement of this problemmay be taken from Pedroni's paper in this volume, which is based on earlierwork by him (Pedroni (1995, 1997a)). A related paper is by Kao (1999)which contains very similar and related analysis. The second route is to takeas the null that of cointegration and is the basis of the test proposed byMcCoskey and Kao (1998a).4 This too is a residual-based test and has as itsanalogue in the time series literature the tests of Harris and Inder (1994),Shin (1994), Leybourne and McCabe (1994) and Kwiatowski et al. (1992).We discuss both approaches in turn. As a preface, it should be noted that theasymptotic analysis of both approaches involves the use of sequential limitarguments. This involves allowing the time series dimension T to growlarge ®rst and then letting N !1.

3.1. Pedroni

The details of the Pedroni method are described in some detail in his paperin this volume. We shall therefore discuss only the key features of theanalysis which are of importance in a more general context. The methodutilizes the residuals from the cointegrating regression given by

yi, t � ái � äi t � x9i, tâi � ei, t, t � 1, 2, . . . , T ; i � 1, . . . , N ; (23)

where âi � (â1i, â2i, . . . , âMi)9, xi, t � (x1i, t, x2i, t, . . . , xMi, t)9. This formu-lation therefore allows for considerable heterogeneity in the panel, sinceheterogeneous slope coef®cients, ®xed effects and individual speci®cdeterministic trends are all permitted.

Under H0, de®ning zi, t � (yi, t, x9i, t)9, î9i, t � (î yi, t, î

x9i, t),

zi, t � zi, tÿ1 � îi, t, (24)

where the process î9i, t satis®es

1��Tp

X[Tr]

t�1

îi, t ) Bi(Ùi) for each i as T !1:

4In a similar vein, Hadri (1998) develops a test of the null of trend versus differencestationarity.



Bi(Ùi) is a vector Brownian motion with asymptotic covariance given by Ùi

(partitioned conformably) with Ù22i assumed to be positive de®nite (to ruleout cointegration amongst the regressors5). The Bi(Ùi) are taken to bede®ned on the same probability space for all i and moreoverE(îi, tî9j,s) � 0 for all i 6� j and for all s, t.

The speci®cation of the process for îi, t therefore imposes cross-sectionalindependence (excepting any common aggregate disturbances) but allowsfor a wide range of temporal dependence in the data. In particular, noexogeneity requirements are imposed on the regressors in (23).

Under these assumptions, Pedroni discusses the construction of sevenpanel cointegration statistics, four based on what he describes as poolingalong the within-dimension and three based on pooling along the between-dimension. Within the ®rst category, three of the four tests involve the useof non-parametric corrections familiar from the work of Phillips and Perron(1988). The fourth is a parametric ADF-based test. In the second category,two of the three tests use non-parametric corrections while the third is againan ADF-based test. If we denote by ãi the autoregressive coef®cient of theresiduals in the ith unit or cross-section, then the ®rst category of tests usesthe following speci®cation of null and alternative hypotheses:

H0 : ãi � 1, for all i, H A : ãi � ã, 1 for all i: (25)

The second category uses

H0 : ãi � 1, for all i, H A : ãi , 1 for all i: (26)

The analogy with the Levin and Lin (1993) and Im et al. (1997) papers isevident, in terms of the heterogeneity permitted under the alternativehypothesis, in the former case for the root of the raw time series, theautoregressive coef®cient in the estimated residuals for the latter.

We next describe the construction and use of the second of the within-dimension tests, called the Panel r-Statistic, referring the reader to Pedroni'spaper for the remaining tests. The non-parametric tests require estimatingÙi and the long-run variance of ui, t, where

ei, t � ãi ei, tÿ1 � ui, t (27)

and ei, t are the residuals from the panel cointegrating regression (23). Theparametric tests estimate

ei, t � ãi ei, tÿ1 �Xk i

k�1

ãikÄei, tÿk � u�i, t (28)

and use the residuals u�i, t to estimate their (simple) variance since the ADFprocedure in (28) whitens the u�i, t.

5The interesting consequences of relaxing this assumption are an important part of the Hall etal. paper.

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The following steps are implemented for the construction of the Panel r-Statistic. The panel cointegrating regression (23) is estimated and theresiduals extracted. Next, the differenced regression given by

Äyi, t � â9iÄxi, t � çi, t, t � 1, 2, . . . , T ; i � 1, . . . , N (29)

is estimated. The residuals çi, t are used to compute a consistent estimator ofÙi, denoted Ùi, such as the Newey-West estimator. From Ùi extract

L211i � Ù11i ÿ Ù21iÙ

ÿ122iÙ921i: (30)

Use ui, t in (27) to compute its long-run variance ó 2i and ëi � 1

2(ó 2

i ÿ s2i ),

where s2i is the simple variance (i.e. ignoring cross-correlations) of ui, t. The

Panel r-Statistic is then given by

T��Np

ZrN ,Tÿ1 � T��Np XN

i�1

XT

t�1

Lÿ211i e

2i, tÿ1

!ÿ1XN

i�1

XT

t�1

Lÿ211i(ei, tÿ1Äei, tÿ1 ÿ ëi):

(31)

In order to de®ne the statistic suitable for making inference, a standardiza-tion based on the moments of the vector of Brownian motion functionals isagain required. Letting V and W to be mutually independent standardBrownian motion processes of dimension 1 and M respectively, de®ne

~â ��

WW 9

� �ÿ1�WV ; (32)

Q � V ÿ ~â9W : (33)

The vector of Brownian motion functionals is then given by

�9 ��

Q2,

�Q dQ, ~â9~â

� �: (34)

Let È denote the vector of means of these functionals, given by

È9 � (È1, È2, È3): (35)

Ø is the variance±covariance matrix of �, such that Ø( j), j � 1, 2, 3 refersto the j 3 j upper sub-matrix of Ø and de®ne ö9(2) � (ÿÈÿ1

1 , È2Èÿ21 ).

Pedroni shows that under H0,

T��Np

ZrN ,Tÿ1 ÿÈ2Èÿ11

��Np) N (0, ö9(2)Ø(2)ö(2)) (36)

and a ®nal standardization provides the standard normal form. The statisticdiverges to negative in®nity under the alternative hypothesis, therebyproviding a consistent test. Large negative values lead to the rejection of thenull hypothesis of no cointegration.

The logic of this exercise, ®rst encountered in the consideration of the



Levin and Lin estimators, generalizes to all seven of the Pedroni estimators,so that

êN ,T ÿ ì��Np��

íp ) N (0, 1) (37)

for the appropriately standardized statistic and corrections for mean andvariance. The correction factors ì and í depend on (a) the statistic consid-ered; (b) the dimension of M ; and (c) whether or not unit-speci®c constantsand/or trends are included in the regression. They are computed by MonteCarlo simulation and are tabulated in easily readable form by Pedroni inTable 2 of his paper.

We turn now to considering the LM-test for the null of cointegration, dueto McCoskey and Kao (1998a).

3.2. McCoskey and Kao

In order to understand this test, consider (23) with äi set to 0.6 Theformulation adopted by McCoskey and Kao (1998a) is to consider ei, t to becomposed of two separate terms,

ei, t � èXt

j�1

ui, j � ui, t: (38)

The regressors are generated as

xi, t � xi, tÿ1 � ùi, t, (39)

where xi, t is M-dimensional as before. Under the null hypothesisH0 : è � 0, (23) is a system of cointegrated regressors. Independence acrosscross-sectional units is maintained, as is the assumption of no cointegrationamongst the regressors. The long-run variance-covariance matrix7 ofwi, t � (ui, t, ù9i, t)9 is next de®ned as

6The framework can be generalized to include unit-speci®c trend terms äi. It involvescomputing mean and variance correction terms appropriate to this case. These corrections areused by McCoskey and Kao in their contribution to this volume but only for the speci®cationsrelevant to their empirical exercise.

7A critical assumption, in contrast with the Pedroni analysis, is of a constant variance±covariance matrix across the cross-sectional units, so that

1��Tp

X[Tr]

t�1

wi, t ) B(Ù):

As noted by McCoskey and Kao (1998a) the analysis can be generalized to allow for this speci®c formof heterogeneity also but, allied to the assumption of independence across i, the relaxation of thisrestriction will imply that the method will be equivalent to equation-by-equation estimation of thesystem.

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Ù � $21 $12

$21 Ù22

� �: (40)

The LM-statistic is given by

LM �1N

XN

i�1

1T 2

XT

t�1

Sy2i, t

$21:2

(41)

where $21:2 is a consistent estimator of

$21:2 � $2

1 ÿ$12Ùÿ122 $21, yy i,k � yi,k ÿ $12Ù

ÿ122 ùi,k

and

Sy i, t �Xt

k�1

(yy i,k ÿ ái ÿ cây i9xi,k):

cây i is the fully-modi®ed estimator (FM) of âi.The construction of this statistic therefore requires a consistent estimate

of Ù in order to implement the non-parametric corrections. By implement-ing these corrections, the FM-estimator is able to take account of serialcorrelation of the residuals in (23) and the endogeneity of the regressors andprovides an estimator which is asymptotically unbiased. A description ofthe method for constructing the FM-estimator, when âi � â for all i, isgiven by Kao et al. in this volume and is therefore not repeated here. Thegeneralization to heterogeneous âi's requires construction of these FM-estimators unit by unit (which may be successfully achieved for largeenough T ) but the principles involved are the same.

The development of the FM-estimator, for the case of homogeneousslope coef®cients, is due to Pedroni (1996) and later by Kao and Chiang(1998) and Phillips and Moon (1999). In the time series context, FM-estimators were ®rst developed by Phillips and Hansen (1990). Kao andChiang (1998) also develop results for the Stock and Watson (1992) class ofdynamic ordinary least squares (DOLS) estimators for panels, where leadsof Äxi, t are included in the cointegrating regressions to achieve the sameeffect as that of full modi®cation in providing asymptotically unbiasedestimators. Since the estimators are asymptotically equivalent, the relativemerits of these two methods therefore boil down to a comparison of theirperformance in ®nite samples. Kao and Chiang (1998) investigate the ®nitesample properties of these estimators and Kao et al. use both methods toinvestigate international R&D spillovers.

To implement their test, McCoskey and Kao de®ne the adjusted LM-statistic as



LMy ��Np

(LM ÿ ìv)

óv: (42)

Under H0 : è � 0,

LMy ) N (0, 1) (43)

and since the statistic diverges under the alternative hypothesis, large valuesof LM y imply rejections of the null hypothesis. The correction factors ìv

and ó 2v are the mean and variance of a complex functional of a Brownian

motion process de®ned in Harris and Inder (1994) but depend only on thedimension of M and on whether or not unit-speci®c constants or trends areincluded in (23).

All this is of course very much in line with the general principles forconstructing such tests that we have tried to describe in this paper. Forexample, the McCoskey and Kao (1998a) test is in the spirit of the Im et al.(1997) analysis, since it involves the averaging of the individual LM-statistics across the cross-sections. They also involve the use of non-parametric corrections (common to the spirit of the Pedroni tests) and meanand variance correction factors (common to all the testing and estimationprocedures described here, except for the Maddala and Wu tests). Thus it ispossible to see many unifying strategic features in this already quite diverseliterature and this is helpful for an understanding of the issues resolved andwhere the important extensions may lie.

Before proceeding to a discussion of these extensions, it is necessary tonote brie¯y the results of McCoskey and Kao and by Kao et al. in thisvolume since both papers make extensive use of the techniques discussed inthis section. McCoskey and Kao use their LM-test to investigate therelationship between urbanization, output per worker and capital per workerin a sample of 30 developing countries and 22 developed countries. They®nd that the existence of a long-run relationship can be sustained, althoughthe sign and magnitude of the impact of urbanization varies widely acrosscountries. Their results are sensitive to whether or not a trend is included inthe cointegrating regressions, to proxy for the growth rate in gross domesticproduct per worker caused by factors other than urbanization and capital perworker. An interesting interpretation of the signi®cant interaction betweenurbanization and time trend is suggested. Under this interpretation, thereappears to be some evidence for urbanization acting as a potential levellerof growth rates across countries. For example, the results show thaturbanization acts as a draining in¯uence on otherwise positive growth whilepropping up growth rates in struggling economies.

Kao et al. return to a proposition put forward by Coe and Helpman(1995), using a sample of 21 OECD countries and Israel, to show that bothdomestic and foreign R&D capital stocks have important effects on the totalfactor productivity (TFP) of countries. Using FM and DOLS methods, Kaoet al. conclude that while evidence of the linkage between domestic capital

622 BULLETIN


stock and TFP is strong, the link between international capital stock andTFP appears to be considerably weaker. They therefore reject an importantconclusion of the Coe and Helpman analysis, namely that internationalR&D spillovers are trade related.

IV. NEW DIRECTIONS

The most comprehensive formalization of the asymptotic theory of non-stationary panel data is due to Phillips and Moon (1999a). The signi®cantaspects of this research are ®rst summarized below. We then move on todiscuss some issues which remain largely unresolved.

4.1. Phillips and Moon

Several important areas have been covered by Phillips and Moon in theirrecent papers to which reference should be made for the details. The ®rst setof signi®cant results is their formalization of the multi-indexed asymptotictheory relevant to panel data. This is based upon distinguishing among threekinds of limiting behaviour of N and T, as described in the introductionabove, and upon deriving conditions where sequential limits are equivalentto joint limits, both for convergence in probability and convergence indistribution. The essential condition is one of uniform convergence (that isconvergence of the random variable is uniform in the N dimension asT !1) for which Phillips and Moon (1999a) present generalized condi-tions applicable to multi-indexed asymptotics.

The second set of important ®ndings relates to looking at the asymptoticbehaviour of estimators in (a) panel-spurious regressions where there is notime series cointegration; (b) heterogeneous panel cointegration, where eachunit has its own cointegrating relation (as in (23) above); (c) homogeneouspanel cointegration (âi � â for all i); and (d) near homogeneous panelcointegration, where the divergence from homogeneity across units is smalland is caused by the value of a so-called localizing parameter. They show inparticular that both for cases (a) and (b) above, it is possible to obtain

��Np

consistent estimates for the long-run average coef®cient using pooled data.8

The limit distributions are demonstrated to be Gaussian. This is a verynotable departure from the analysis of Pesaran and Smith (1995) whoestablished that average effects would not be consistently estimated frompooled data if the panel were characterized by heterogeneous cointegratingrelationships across the different units. This departure arises from a some-what different and `robust' de®nition of the long-run average coef®cient interms of the long-run average variance of the panel which can exist even inthe absence of any cointegration.

For cases (c) and (d), Phillips and Moon derive pooled fully-modi®ed

8See also Kao (1999).



estimators which are T��Np

consistent and also have Gaussian limitingdistributions. Use can be made of these results to formulate hypothesis testsabout the long-run average parameters between subsets of the sample.

In their contribution to this volume, Moon and Phillips develop somelocal-to-unity asymptotics for maximum likelihood estimation in dynamicpanels. While the local-to-unity parameter can be estimated consistently inthe absence of deterministic trends or in their presence but in homogeneousform, heterogeneous deterministic trends lead to the maximum likelihoodestimator being inconsistent. The inconsistency arises from the number ofincidental parameters (namely the deterministic trends) going to in®nity asN !1. In a range of panel data contexts, such as in the estimation ofdynamic models with ®xed effects, the inconsistency disappears as T !1.Moon and Phillips show that the incidental parameter problem persists intheir framework even when both N !1 and T !1.

4.2. Unresolved Issues and Extensions

Several issues remain which are worthy of further investigation. A substan-tial part of both the theory and the practice in this area has been based onconsidering and using the asymptotic properties of estimators and tests. It isclear that a substantial amount of simulation work is necessary to establishsystemtatically the role of asymptotic theory in estimates and tests derivedfrom ®nite samples.

More signi®cantly, a formal study of the relaxation of the pervasivecross-sectional independence assumption is necessary in order to makethe asymptotic theory relevant to empirical examples. This is because inmany multi-country contexts, with dependence on common globalshocks, the independence assumption is very likely to be violated. In theconclusion to their paper, Phillips and Moon (1999a) discuss the consid-erable dif®culties involved not only in incorporating dependence into theanalysis but also of characterizing it appropriately. However, the authorsof three of the papers published in this volume have made an importantstart in this direction, and two of these papers appear here. Maddala andWu use bootstrapping methods to allow for cross-section dependence.Pedroni (1997b), in his study of purchasing power parity, proposes theuse of GLS-based corrections to allow for feedback across individualmembers of the panels. In his study of violations of purchasing powerparity relations, the corrections for cross-sectional dependence do notappear to play a signi®cant role but this should not be to deny thepotential of this strand of the research agenda. Finally, the paper by Hallet al. introduces a range of ideas which should lead to interesting newresearch.

Building on earlier work by Hall and Urga (1998), the Hall et al. paperprovides another counter-example to the Pesaran and Smith (1995) analysis.Focusing on the structure of the regressors in the panel cointegrating

624 BULLETIN


regressions, Hall et al. demonstrate that if the regressors in each unit of thepanel are driven by common stochastic trends, and each unit cointegrates,then a standard panel data estimator which imposes homogeneous para-meters across the panel will provide consistent estimates of the averagelong-run effects even if the true model has heterogeneous parameters. Thepaper then goes on to develop a new principal-components-based test forcommon stochastic trends and presents size and power properties of thesetests in ®nite samples. The testing methods are consistent even in thepresence of a mixture of I(0) and I(1) variables, thereby obviating the needto pre-test the panel for unit roots.

Two signi®cant features of this work should be noted, particularlybecause they throw light on the tensions in the literature and the develop-ments required. First, by working with common driving trends across theunits of the panel, cross-sectional dependence is introduced explicitly.However, except for proving consistency, the asymptotic theory of theprincipal-component estimators is not developed and the analysis of theproperties relies heavily on the simulations conducted. This may prove tobe a dif®cult task, but the analysis represents an important step in the rightdirection.

The second feature is to allow for multiple cointegrating vectors, albeitonly in the regressor set. This too is a necessary extension of theestimation and inference framework which nevertheless remains veryincomplete in this respect. The cointegrating vector (in each unit of thepanel) is usually taken to be unique and issues of normalization (i.e. to dowith the choice of the regressand) are not addressed. Only Larsson,Lyhagen and LoÈthgren (1998), by proposing the panel data analogue of theJohansen maximum likelihood method, study the case of multiple cointe-grating vectors in panels. The price they pay is their need to maintain theassumption of cross-sectional independence. Their suggested statistic,inspired by the Im et al. (1997) test for unit roots, is the average (acrossunits) of the Johansen likelihood ratio statistic. This is asserted to have astandard normal density once the mean and variance corrections have beenimplemented, where these moments are derived as usual by stochasticsimulations.

The status of the results in Larsson et al. (1998) is unclear since the maintheorem is asserted and not proved. Nevertheless, if correct, it would markimportant progress. An overarching analysis which combines the relaxationof the independence condition by Hall et al. with a consideration of multiplecointegrating vectors due to Larsson et al. (1998) would denote a huge stepforward in the study of integrated dynamic panels.

V. CONCLUSION

This volume contains descriptions and the use of all the major testing andestimation procedures for unit roots and cointegration in heterogeneous



integrated dynamic panels. It is in that sense self-contained.9 The overviewhas drawn attention to many of the important ideas and techniques whichhave emerged in the literature. In particular, we have considered (i) theasymptotic behaviour of panel estimators in the presence of integratedseries, including their rates of convergence, modes of convergence (sequen-tial, limited or uncontrolled) and the forms of the limiting distibutions; (ii)the construction of pooled panel and group-mean tests for unit roots; (iii)the role of ADF regressions vis-aÁ-vis non-parametric corrections, as a wayof allowing for serially dependent and heteroscedastic residual processes;(iv) the construction of tests for cointegration under both speci®cations ofthe null (i.e. no cointegration and cointegration); (v) the use of mean andvariance corrections using moments of Brownian motion functionals com-puted by stochastic simulation; and (vi) full modi®cation to take account ofendogeneity of the regressors. Several important issues remain to be devel-oped in future research in an area which is proving to be richly productive.A better understanding of these issues and generalizations of the techniqueswill help to address the many interesting empirical questions in areas asdiverse as growth, international ®nance and industrial structure. It is in thishope that this collection has been compiled.

Institute of Economics and Statistics, Oxford

Date of Receipt of Final Manuscript: July 1999

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